Computing devices have made significant contributions toward the advancement of modern society and are utilized in a number of applications to achieve advantageous results. Numerous devices, such as digital cameras, computers, game consoles, video equipment, hand-held computing devices, audio devices, telephones, and navigation systems have facilitated increased productivity and reduced costs in communicating and analyzing data in most areas of entertainment, education, business and science. The digital camera and camcorders, for example, have become popular for personal use and for use in business.
FIG. 1 shows an exemplary digital camera. The digital camera 100 typically includes one or more lenses 110, one or more filters 120, one or more image sensor arrays 130, an analog-to-digital converter (ADC) 140, a digital signal processor (DSP) 150 and one or more computing device readable media 160. The image sensor 130 includes a two-dimension array of hundreds, thousand, millions or more of imaging pixels, which each convert light (e.g. photons) into electrons. The image sensor 130 may be a charge coupled device (CCD), complementary metal oxide semiconductor (CMOS) device, or the like. The filter 120 may be a Bayer filter that generates a mosaic of monochrome pixels. The mosaic of monochrome pixels are typically arranged in a pattern of red, green and blue pixels.
In an exemplary implementation, the digital camera 100 may include a lens 110 to focus light to pass through the Bayer filter 120 and onto the image sensor 130. The photons passing through each monochrome pixel of the Bayer filter 120 are sensed by a corresponding pixel sensor in the image sensor 130. The analog-to-digital converter (ADC) 140 converts the intensity of photons sensed by the pixel sensor array into corresponding digital pixel data. The raw pixel data is processed by the DSP 150 using a demosaic algorithm to produce final interpolated pixel data. The final interpolated pixel data is typically stored in one or more of the computing device readable media 160. One or more of the computing device readable media 160 may also store the raw pixel data.
Image denoising is an important part of image enhancement of digital and analog image acquisition systems. Image denoising has been a widely studied problem in image processing. Among the techniques, wavelet domain denoising has received considerable interest. Wavelets are a theory and algorithms to achieve a linear space-frequency decomposition of a signal. Referring now to FIGS. 2A and 2B an exemplary image and corresponding wavelet transform is shown. Wavelets permit the representation of a signal as a sum of dyadically scaled and translated copies of a “mother” wavelet. For each resolution (e.g., scale), starting from the highest to the coarsest, the signal is iteratively decomposed into a low-pass sub-band and several high-pass residual sub-bands (one for each orientation) resulting in a multi-scale sub-band decomposition of the signal, as illustrated in FIG. 2B. High-pass coefficients encode the missing information to be able to reconstruct the high resolution signal in each orientation from the low-pass version. Conventional wavelet denoising techniques include wavelet hard and soft thresholding and multivariate or spatially adaptive extension.
Noise herein refers to random degradation present in images as opposed to deterministic or nearly deterministic distortions such as optical aberrations. The main sources of noise in digital based image acquisition systems (e.g., charge coupled devices (CCD), complementary metal oxide semiconductor (CMOS) and the like) include, but are not limited to photon noise, dark current noise, readout noise and fixed-pattern noise. Photon noise (e.g., shot noise) refers to the inherent natural variation in the process of counting the photon in the incident light flux. The number of photon arrivals during a given recording period is governed by Poisson statistics whose variance is equal to the average total number of photons. Dark noise (e.g., thermal noise) comes from the detection of thermally generated electrons within the sensor (e.g., CCD material). Dark current describes the rate of the generation of thermal electrons at a given sensor temperature. Readout noise refers to the uncertainty introduced during the process of quantifying the electronic signal on the chip. The major component of readout noise comes from the on-chip preamplifier. Some authors make the distinction between on-chip electronic noise and amplifier noise but readout noise is most often defined as the sum of all camera-characteristics signal-independent noises. Fixed pattern noise (FPN) is due to differences in individual sensor cell (e.g., pixel) sensitivity. This type of noise is more visible at higher intensities and is signal-dependent. It statistics are proportional to the original signal.
Image restoration is typically based on the modeling of both the expected image signal and the degradation such as the contaminating noise. Removal of noise from images relies on differences in the statistical properties of noise and expected signal as the artifacts arising from many imaging devices is quite different from the image that the noise contaminates. Using models describing signal and noise statistics, various estimation techniques are used to reduce the noise in the observed image signal. However, due to the high dimensionality (e.g., long range) in spatial correlations of natural images, modeling the statistics of these images is complex. In particular, there is a high, long-range dependency between pixels in the same neighborhood. For example, pixels belonging to the same texture are correlated and the range of this correlation can go well beyond the short range of a few pixels in regular images. To address the issue, a few assumptions are commonly made to simplify the signal models.
A first assumption is the stationarity also referred to as translation-invariance in image processing. In translation invariance the statistical distribution of pixels in a given neighborhood is the same for all locations within the image. A second assumption is that the autocorrelation between any pair of neighboring pixels is rapidly decaying with the distance between the pixels. A third common assumption is that the statistical distribution of pixels in a neighborhood (signal, noise or both) follows univariate or multivariate Gaussian statistics. These assumptions are not all suitable for natural images that commonly contain a variety of large correlated structures.
For an image contaminated by additive white Gaussian noise (AWGN) the image can be expressed by:y=x+n  (1)Where y is the noisy observation, x the noise-free signal to be estimated and n the AWGN. If the noise is AWGN in the spatial (e.g., regular) domain, the noise is AWGN in the wavelet domain too because the wavelet transform is linear and orthogonal. For multivariate probability distributions: P(A|B) is the conditional probability of event A given B; P(A, B) is the joint probability of having both events A and B together; and P(A) is the marginal probability of event A regardless of B.
In the context of the minimum mean square error (MMSE) estimation, the mean of the posterior distribution provides an unbiased estimate of the variable x, given measurement y. MMSE takes a weighted average:x′(y)=Px|y(x|y)×dx  (2)Applying Bayes' formula, Px|y(x|y)Py(y)=Px,y(x,y)=Px|y(x|y)Px(x) the probability densities of the noise and signal:x′(y)=∫xPy|x(y|x)Px(x)×dx/Py(y)  (3)andPy(y)=∫xPy|x(y|x)Px(x)dx  (4)Py|x is actually the probability density of the noise, which can be rewritten as Pn(n)=Pn(y−x) and Px is the prior probability density of the signal, which gives:x′(y)=∫xPn(y−x)Px(x)×dx/∫xPn(y−x)Px(x)dx  (5)The denominator is the probability density function (PDF) of the noisy observation—convolution of the noise and the signal's PDFs.
In one technique, a univariate generalized Gaussian distribution (GGD) model, which closely fits the distribution of wavelet coefficients in natural images, has been extended with a multivariate (GGD). The technique estimates the MGGD core parameters using a set of “training images.” The signal covariance matrix is estimated from the noisy observation under the additive white Gaussian noise (AWGN) assumption. The model is used to resolve a Bayesian maximum posteriori (MAP) estimation equation. The technique takes approximately 15-45 second for each channel of a 512×512 image with A Daubechies 8 wavelet filter on a Pentium IV type personal computer.
In another technique, a multivariate model for discrete wavelet transform (DWT) coefficient neighborhood using scale mixtures of Gaussian distribution is developed. With this model and under the additive Gaussian noise of know covariance assumption, a Bayesian least squares estimated is constructed. This Gaussian scale mixture (GSM) model can represent the leptokurtotic behavior of the wavelet coefficient distributions, as well as the dependence of their local amplitudes. The mixture of Gaussian leads to mathematically more tractable solutions than the MGGD. The technique takes about 20-60 second for each channel of a 256×256 image on a Pentium III type personal computer.
The conventional techniques are computationally intensive. The conventional techniques may also cause blurring, smoothing or the like of the image particular around edges, lines and texture areas or the like. Accordingly, what is needed is a computationally less expensive noise correction technique. What is also needed is a denoising technique providing an improved noise correction quality.